Abstract

We introduce a $q,t$-enumeration of Dyck paths that are forced to touch the main diagonal
at specific points and forbidden to touch elsewhere
and conjecture that it describes the action of
the Macdonald theory $\nabla$ operator applied to a Hall--Littlewood
polynomial. Our conjecture refines several earlier conjectures concerning
the space of diagonal harmonics including the ``shuffle conjecture"
(Duke J. Math. $\mathbf {126}$ ($2005$), 195-232) for $\nabla e_n[X]$.
We bring to light that certain generalized Hall--Littlewood polynomials
indexed by compositions are the building blocks for the algebraic
combinatorial theory of $q,t$-Catalan sequences, and we prove a number of
identities involving these functions.